What makes sense to call a quotient (as sets) of the 2-adic numbers in which every element of a given equivalent class shares the same final convergent subsequence, e.g. $-\frac13\sim-\frac23\sim\frac13\ldots$?
For example $\overline{01}_2$ is a subsequence of $\overline{10}_2$ and vice versa, since either can be truncated to arrive at the other and therefore they are equivalent.
Is there any reason why that's not well-defined?
Define these classes by the graphing the orbit of the truncation function. Then you can say two numbers are equivalent if their graph is connected.
Is such an object well-studied?
Following the comments it is starting to look like say $\mathcal C/\overline{\mathcal C}$ where $\overline {\mathcal C}$ is the endpoints of the removed segments of the Cantor set.
The OP meant $\sum_{n\ge -N} a_n p^n \sim \sum_{m\ge -M} b_m p^m$ iff there is $k,l$ such that $a_n=b_{n+k}$ for all $n\ge l$.
Why didn't the OP phrase it correctly?
$\sum_{n\ge -N} a_n p^n \sim 1+\sum_{n\ge -N} a_n p^n$ iff $\exists n\ge 0, a_n \ne p-1$.
So there is a special case for the negative integers --> not good.
Removing this special case, it becomes $a\sim b$ iff $\exists d\in \Bbb{Z},c\in \Bbb{Z}[p^{-1}]$, $a=p^d b+c$, which is the quotient by the action of a group.